Question

In: Statistics and Probability

Many regions in North and South Carolina and Georgia have experienced rapid population growth over the...

Many regions in North and South Carolina and Georgia have experienced rapid population growth over the last 10 years. It is expected that the growth will continue over the next 10 years. This has motivated many of the large grocery store chains to build new stores in the region. The Kelley’s Super Grocery Stores Inc. chain is no exception. The director of planning for Kelley’s Super Grocery Stores wants to study adding more stores in this region. He believes there are two main factors that indicate the amount families spend on groceries. The first is their income and the other is the number of people in the family. The director gathered the following sample information.

Family Food Income Size
1 $ 5.26 $ 73.98 5
2 4.08 54.90 2
3 5.76 59.12 4
4 3.48 52.02 1
5 4.20 65.70 2
6 4.80 53.64 4
7 4.32 79.74 3
8 5.04 68.58 4
9 6.12 165.60 5
10 3.24 64.80 1
11 4.80 138.42 3
12 3.24 125.82 1
13 6.45 77.58 7
14 4.63 172.92 6
15 6.60 90.98 8
16 5.40 141.30 3
17 6.00 36.90 5
18 5.40 56.88 4
19 3.36 71.82 1
20 4.68 69.48 3
21 4.32 54.36 2
22 5.52 87.66 5
23 4.56 38.16 3
24 5.40 43.74 7
25 4.78 60.72 4

Food and income are reported in thousands of dollars per year, and the variable size refers to the number of people in the household.

  Click here for the Excel Data File

  1. a-1. Develop a correlation matrix. (Round your answers to 3 decimal places. Negative amounts should be indicated by a minus sign.)

  1. a-2. Do you see any problem with multicollinearity?

  1. b-1. Determine the regression equation. (Round your answer to 3 decimal places.)

  1. b-2. How much does an additional family member add to the amount spent on food? (Round your answer to the nearest dollar amount.)

  1. c-1. What is the value of R2? (Round your answer to 3 decimal places.)

  1. c-2. Complete the ANOVA (Leave no cells blank - be certain to enter "0" wherever required. Round SS, MS to 4 decimal places and F to 2 decimal places.)

  1. c-3. State the decision rule for 0.05 significance level. H0: = β1 = β2 = 0; H1: Not all βi's = 0. (Round your answer to 2 decimal places.)

  1. c-4. Can we reject H0: = β1 = β2 = 0?

  1. d-1. Complete the table given below. (Leave no cells blank - be certain to enter "0" wherever required. Round Coefficient, SE Coefficient, P to 4 decimal places and T to 2 decimal places.)

  1. d-2. Would you consider deleting either of the independent variables?

  1. State true or false.

From the graph the residuals appear normally distributed.

  • True

  • False

  1. Choose the right option from the following graph.
  • There is a homoscedasticity problem.

  • There is no homoscedasticity problem.

Solutions

Expert Solution

Solution

a-1

Food Income Size
Food 1.000
Income 0.107 1.000
Size 0.867 0.132 1.000

a-2. Do you see any problem with multicollinearity?

No Since the correlation between Income and size < 0.5.

b-1. Determine the regression equation.

we will solve it by using excel and the steps are

Enter the Data into excel

Click on Data tab

Click on Data Analysis

Select Regression

Select input Y Range as Range of dependent variable.

Select Input X Range as Range of independent variable

click on labels if your selecting data with labels

click on ok.

So this is the output of Regression in Excel.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.8668
R Square 0.7514
Adjusted R Square 0.7288
Standard Error 0.4983
Observations 25.0000
ANOVA
df SS MS F Significance F
Regression 2.0000 16.5115 8.2558 33.2446 0.0000
Residual 22.0000 5.4633 0.2483
Total 24.0000 21.9749
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 3.3044 0.2875 11.4923 0.0000 2.7081 3.9007 2.7081 3.9007
Income -0.0002 0.0027 -0.0733 0.9422 -0.0058 0.0054 -0.0058 0.0054
Size 0.4218 0.0521 8.0922 0.0000 0.3137 0.5298 0.3137 0.5298

Food = 3.304 - 0.000*Income+0.422*Size

b-2. How much does an additional family member add to the amount spent on food?

$0.4218

c-1. What is the value of R2?

R2=0.7514

c-2. Complete the ANOVA

ANOVA
df SS MS F Significance F
Regression 2.0000 16.5115 8.2558 33.24 0.0000
Residual 22.0000 5.4633 0.2483
Total 24.0000 21.9749

c-3. State the decision rule for 0.05 significance level. H0: = β1 = β2 = 0; H1: Not all βi's = 0.

If P-value < 0.05 significance level reject the null hypothesis.

c-4. Can we reject H0: = β1 = β2 = 0?

Yes, we reject H0: = β1 = β2 = 0

d-1. Complete the table given below.

Coefficients Standard Error t Stat P-value
Intercept 3.3044 0.2875 11.49 0.0000
Income -0.0002 0.0027 -0.07 0.9422
Size 0.4218 0.0521 8.09 0.0000

Would you consider deleting either of the independent variables?

Income variable should be deleted.

  1. State true or false.

From the graph the residuals appear normally distributed.

  • True

  1. Choose the right option from the following graph.

  • There is no homoscedasticity problem


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